
theorem Th8:
  for X,Y,Z being non empty TopSpace for f being continuous
  Function of Y,Z holds oContMaps(X, f) is monotone
proof
  let X,Y,Z be non empty TopSpace;
  let f be continuous Function of Y,Z;
  let a,b be Element of oContMaps(X, Y);
  the TopStruct of Y = the TopStruct of Omega Y & the TopStruct of Z = the
  TopStruct of Omega Z by WAYBEL25:def 2;
  then reconsider f9 = f as continuous Function of Omega Y, Omega Z by
YELLOW12:36;
  reconsider g1 = a, g2 = b as continuous Function of X, Omega Y by Th1;
  set Xf = oContMaps(X, f);
  reconsider fg1 = f9*g1, fg2 = f9*g2 as Function of X, Omega Z;
  g2 is continuous Function of X,Y by Th2;
  then
A1: Xf.b = f9*g2 by Def2;
  assume a <= b;
  then
A2: g1 <= g2 by Th3;
  now
    let x be set;
    assume x in the carrier of X;
    then reconsider x9 = x as Element of X;
A3: (f9*g2).x9 = f9.(g2.x9) by FUNCT_2:15;
    ( ex a, b being Element of Omega Y st a = g1.x9 & b = g2.x9 & a <= b)
    & (f9*g1) .x9 = f9.(g1.x9) by A2,FUNCT_2:15;
    then (f9*g1).x9 <= (f9*g2).x9 by A3,WAYBEL_1:def 2;
    hence ex a,b being Element of Omega Z st a = (f9*g1).x & b = (f9*g2).x & a
    <= b;
  end;
  then
A4: fg1 <= fg2;
  g1 is continuous Function of X,Y by Th2;
  then Xf.a = f9*g1 by Def2;
  hence Xf.a <= Xf.b by A1,A4,Th3;
end;
